190 XIII INTEGRATION
13.9, Problems 13 and 24). Show that A(A) is either 0 or 1. (Suppose that d = A(A) e ]0,1 [.
Show that, for each component interval J of I — En, we have
A(A n J) ^ (2dl(l 4- <0)A(J).
For this purpose use (a) above and Problem l(c) of Section 13.21 to show that
dt r1 , v dt rd dt
for suitably chosen integers /?, q, r, s1 with /w — qr = ± 1. Use Problem 2 to complete
the proof.)
(c) Show that the measure p. — g • A, where g(x) = !/(!+ x), is invariant with respect
to p. Deduce from (b) and the ergodic theorem (Section 13.9, Problems 13 and 24)
that, for every A-integrable function / on I and almost all irrational x e I, we have
n~*oo n log 2 JQ 1 -f t
(Gauss-Kuzmin formula).
(d) Deduce from (c) that, for each integer p ^ 1, if vn(x, p) is the number of indices
k<^n such that qk(x) —p(x irrational), then
log
lOg 2
for almost all irrational x e I.
(e) Deduce from (c) that
oo / 1 \ log n/log 2
lim fofrte&c) • • -^W)1'" = FT 11 + -TTV
n-»« n = i \ n2 + 2n]
for almost all irrational x.
15. THE LEBESGUE-NIKODYM THEOREM AND THE ORDER RELATION
ON MR(X)
(13.15.1) Let \JL and v be two positive measures on a locally compact space X,
such that v ^ ju. Then there exists a locally n-integrable function g such that
We distinguish two cases.
(I) Suppose first that v is bounded. Since the function 1 is then v-integ-
rable, it follows from (1 3.1 1 .2.2) that for every /e Jf R(X)
(13.15.1.1) Deduce from (a) that if r=*p/q and r' — p'lq' are two rational numbers belonging